How classification neural nets are different from simple dimension reduction + clustering?
PCA is linear, NN are nonlinear, more generally it's a universal function approximator.
That said basic NN are not terribly usefull, the real value of NN is for structured data which is obvious for us but hard to describe analytically. Basically NN architecture has been designed to learn this structure using operations like convolution and maxpooling.
There are still problems of course, and in some ways I think clustering algorithms still have value for things like anomaly detection.